Emergence of spatial structure (stripes, spots, hexagons, spirals) in spatially extended non-equilibrium systems via linear instabilities of a homogeneous base state. Core paradigms: Turing 1952 reaction-diffusion (activator-inhibitor…
pattern-formation
Swift-Hohenberg 1977 normal form for pattern onset
Swift-Hohenberg 1977 (Phys. Rev. A 15:319) proposed a minimal normal-form PDE capturing the essentials of Rayleigh-Bénard convection onset…
Real Ginzburg-Landau amplitude equation near pattern onset
Real Ginzburg-Landau equation derived by Newell-Whitehead-Segel 1969 for pattern onset in spatially-extended non-equilibrium systems. …
Fisher-KPP 1937 travelling wave with selected minimum speed
Fisher 1937 (Ann. Eugenics 7:355) and Kolmogorov-Petrovskii-Piskunov 1937 (Bull. Univ. Moscou A 1:1) independently introduced the 1D…
SH dispersion σ(k) = ε - (k² - k_0²)²: concave max at k_0, ε_c = 0
Linear stability analysis of the Swift-Hohenberg equation around the homogeneous base state ψ = 0. Linearised growth rate: σ(k) = ε - (k²…
Real GL amplitude: saturation |A|² = ε/g with residual ≡ 0
Steady-state of the real Ginzburg-Landau amplitude equation. Setting Ȧ = εA - gA³ = 0 gives two branches: A = 0 (trivial) and A = ±√(ε/g)…
Fisher-KPP c* = 2√(rD): derived from linearised dispersion discriminant
Derivation of the classical minimum wave speed. Linearise u_t = ru(1-u) + D·u_xx at u = 0 to obtain u_t = ru + D·u_xx; seek…
Turing reaction-diffusion 2x2 Jacobian: characteristic polynomial via Cayley-Hamilton
Turing reaction-diffusion framework via Cayley-Hamilton. Setup: activator-inhibitor system du/dt = f(u, v) + D_u d_xx u, dv/dt = g(u, v) +…
Newell-Whitehead-Segel amplitude equation: A_T = A - |A|^2 A (complex Ginzburg-Landau onset)
Newell-Whitehead-Segel amplitude-equation framework (NWS 1969 J Fluid Mech 38, 279). Setup: weakly nonlinear reaction-diffusion /…
Swift-Hohenberg roll-amplitude saturation: A^2_saturation = 4r/3 (Lyapunov functional minimum)
Swift-Hohenberg roll-amplitude saturation framework. Setup: Swift-Hohenberg equation u_t = r u - (1 + d_xx)^2 u - u^3 (Swift-Hohenberg 1977…
Theorem: Cayley-Hamilton of Turing J gives zero matrix; det(M(43/20)) = -1809/40 < 0 (Turing instability)
Theorem (Cayley-Hamilton Turing-dispersion canonical): for Jacobian J = [[5, -6], [6, -7]] with tr(J) = -2, det(J) = 1, and diffusion D =…
Theorem: NWS amplitude equation A_T = A - |A|^2 A has unique positive steady state A_s = 1
Theorem (NWS amplitude-saturation canonical): the dimensionless Newell-Whitehead-Segel amplitude equation A_T = A - |A|^2 A admits three…
Theorem: Swift-Hohenberg roll-amplitude saturation A^2 = 4r/3 (via Lyapunov variational minimum)
Theorem (Swift-Hohenberg roll-saturation canonical): the Lyapunov functional F[A]/L = -r A^2/4 + 3 A^4/32 (per unit length on q_c = 1…
Turing instability (1952)
A Turing 1952 'Chemical Basis of Morphogenesis'; activator-inhibitor reaction-diffusion patterns; modern animal-coat-pattern Murray +…
RB convection (Rayleigh 1916)
Lord Rayleigh 1916 Ra critical = 1708 for top/bottom isothermal; modern soft-bound + power-law scaling Grossmann-Lohse 2000.
Swift-Hohenberg (1977)
J Swift-P Hohenberg 1977 amplitude-equation for stripe-patterns; modern weakly-nonlinear theory of convection + chemical-patterns.
Ginzburg-Landau
Ginzburg-Landau 1950 amplitude-equation generic for symmetry-breaking; modern complex-GL Aranson-Kramer 2002 review.
Cross-Hohenberg review (1993)
M Cross-P Hohenberg 1993 'Pattern formation outside of equilibrium' Rev Mod Phys; foundational comprehensive review.
Morphogenetic (Wolpert 1969)
L Wolpert 1969 positional-information + French-flag; modern signaling-gradients + reaction-diffusion synthesis (Sharpe 2019).
Turing detail (1952)
A Turing 1952 reaction-diffusion-morphogenesis; modern modern foundational + 2024 cell-fate ML-deep-learning verifications zebrafish-fish.
Bénard convection (1900)
H Bénard 1900 + Rayleigh 1916 + Lorenz 1963 convective-instability; modern modern foundational + Pearson 1958 surface-tension Marangoni.
Swift-Hohenberg (1977)
J Swift-P Hohenberg 1977 SH-equation; modern modern foundational text + amplitude-equations + Cross-Hohenberg 1993 review.
Cahn-Hilliard (1958)
J Cahn-J Hilliard 1958 + Cahn 1961 spinodal-decomposition; modern modern foundational text + 2024 ML-coarse-graining diffusion materials.
DLA (Witten-Sander 1981)
T Witten-L Sander 1981 diffusion-limited-aggregation; modern modern foundational text + 2024 fractal-microbial-colonies + drainage networks.
Keller-Segel (1970)
E Keller-L Segel 1970 chemotaxis-aggregation; modern modern foundational text + Dictyostelium + 2024 collective-cell-migration cancer.